Precalculus Examples

$\frac{x}{x + 1} - x^{3}$

Step 1

To write $- x^{3}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x}{x + 1} - x^{3} \cdot \frac{x + 1}{x + 1}$

Step 2

Simplify terms.

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Step 2.1

Combine $- x^{3}$ and $\frac{x + 1}{x + 1}$ .

$\frac{x}{x + 1} + \frac{- x^{3} (x + 1)}{x + 1}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{x - x^{3} (x + 1)}{x + 1}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $x$ out of $x - x^{3} (x + 1)$ .

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Step 3.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} - x^{3} (x + 1)}{x + 1}$

Step 3.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 - x^{3} (x + 1)}{x + 1}$

Step 3.1.3

Factor $x$ out of $- x^{3} (x + 1)$ .

$\frac{x \cdot 1 + x (- x^{2} (x + 1))}{x + 1}$

Step 3.1.4

Factor $x$ out of $x \cdot 1 + x (- x^{2} (x + 1))$ .

$\frac{x (1 - x^{2} (x + 1))}{x + 1}$

Step 3.2

Apply the distributive property.

$\frac{x (1 - x^{2} x - x^{2} \cdot 1)}{x + 1}$

Step 3.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.3.1

Move $x$ .

$\frac{x (1 - (x \cdot x^{2}) - x^{2} \cdot 1)}{x + 1}$

Step 3.3.2

Multiply $x$ by $x^{2}$ .

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Step 3.3.2.1

Raise $x$ to the power of $1$ .

$\frac{x (1 - (x^{1} x^{2}) - x^{2} \cdot 1)}{x + 1}$

Step 3.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x (1 - x^{1 + 2} - x^{2} \cdot 1)}{x + 1}$

Step 3.3.3

Add $1$ and $2$ .

$\frac{x (1 - x^{3} - x^{2} \cdot 1)}{x + 1}$

Step 3.4

Multiply $- 1$ by $1$ .

$\frac{x (1 - x^{3} - x^{2})}{x + 1}$

Step 3.5

Reorder terms.

$\frac{x (- x^{3} - x^{2} + 1)}{x + 1}$

Step 4

Simplify with factoring out.

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Step 4.1

Factor $- 1$ out of $- x^{3}$ .

$\frac{x (- (x^{3}) - x^{2} + 1)}{x + 1}$

Step 4.2

Factor $- 1$ out of $- x^{2}$ .

$\frac{x (- (x^{3}) - (x^{2}) + 1)}{x + 1}$

Step 4.3

Factor $- 1$ out of $- (x^{3}) - (x^{2})$ .

$\frac{x (- (x^{3} + x^{2}) + 1)}{x + 1}$

Step 4.4

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{x (- (x^{3} + x^{2}) - 1 (- 1))}{x + 1}$

Step 4.5

Factor $- 1$ out of $- (x^{3} + x^{2}) - 1 (- 1)$ .

$\frac{x (- (x^{3} + x^{2} - 1))}{x + 1}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Rewrite $- (x^{3} + x^{2} - 1)$ as $- 1 (x^{3} + x^{2} - 1)$ .

$\frac{x (- 1 (x^{3} + x^{2} - 1))}{x + 1}$

Step 4.6.2

Move the negative in front of the fraction.

$- \frac{x (x^{3} + x^{2} - 1)}{x + 1}$